This came to me in an Old Milwaukee induced trance. Use the current cap and work on a better seal. Once this improved seal is made slowly apply increasing back pressure on outflow pipe to reduce flow as much as possible. Once reduced flow is achieved, resume pumping heavy mud back down the hole, or through side vents of cap. This should give the mud a chance to get a foothold deeper in the hole instead of getting blown out at high pressure. Should be a safe approach not risking doing more damage to the BOP if done properly. It would also leave the current set-up in place which is catching much of the oil.